CS 100 - Week 15 Lecture 1 - 12-4-12
Little bit more from Chapters 9 and 10...
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Aside: additional truth table done after clicker question:
* given the argument:
p->q
~p
.'. q
p qC ~p* p->q*
---------------------
T T F T
T F F F
F (T) (T) (T)
F (F) (T) (T) <---
...but there's a row where the premises are all true
and the conclusion is False -- SO, this argument is thus
shown as NOT valid.
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Back to Chapter 9 - quick review/discussion of converting
categorical statements to one of the 4 standard categorical
forms:
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The 4 standard categorical forms:
* All S are P.
* No S are P.
* Some S are P.
* Some S are not P.
* some tips for converting a statement to
one of these forms (if it's a statement suitable
for that)
Tip 1: REPHRASE a non-standard subject or predicate
so it is a class of things.
Tip 2: REPHRASE a non-is/are verb into is or are
(or is not, are not)
Tip 3: FILL IN any unexpressed quantifiers
(this can be tricky, be charitable)
Tip 4: Translate singular statements to All or
No statements
* see pp. 233-235 for some examples of COMMON
English phrasings that correspond to EACH of these
standard forms
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Back to Chapter 10 -- the same approach holds even with more
variables when using truth tables to show that a deductive
argument expressed in propositional logic form is valid --
...you just have more rows in the truth table!
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* 3 variables? (2 to the 3rd power) = 8 possible
combinations of T and F for those 3 variables;
* 4 variables? (2 to the 4th power) = 16 possible
combinations of T and F for those 4 variables;
...etc.!
* Here's an example with 3 variables:
argument:
p&q
qvr
.'. r
* have a column for each variable to start --
and with 3, "traditional" to have first
have values TTTTFFFF, second to have TTFFTTFF,
and third to have TFTFTFTF -- see the pattern?
* then, as usual, add columns to "build up" to the
premises and conclusion as needed,
mark premises with * and conclusion with C in
header,
and circle premises and conclusion values ONLY in
rows where ALL of the premises are true;
* SO:
p q rC p&q* qvr*
--------------------------
T T (T) (T) (T)
T T (F) (T) (T) <---
T F T F T
T F F F F
F T T F T
F T F F T
F F T F T
F F F F F
...there's a row in which ALL the premises are true
but the conclusion is false, SO this argument IS
NOT valid.
What about:
argument:
p&q
qvr
.'. q
p qC r p&q* qvr*
--------------------------
T (T) T (T) (T)
T (T) F (T) (T)
T F T F T
T F F F F
F T T F T
F T F F T
F F T F T
F F F F F
...here, for ALL rows where BOTH premises are
true, the conclusion is true, SO *this* argument
IS valid.